1

CONTRIBUTION STUDY OF THE THERMODYNAMICS PROPERTIES OF THE

AMMONIA-WATER MIXTURES

by

Sahraoui KHERRIS 1,2,

Mohammed MAKHLOUF

2 Djallel ZEBBAR

1 Omar SEBBANE

3

1Institute of Sciences and Technology,

University of Tissemsilt, B.P. 182 Tissemsilt - Tel. /Fax: 00213 40 74 24 72 (Algeria) 2Reactifs Systems and Materials Laboratory,

Djillali LIABES University, BP 89 Sidi-Bel-Abbes (Algeria) 3Materials and Renewable Energy Laboratory

Abou Bekr Belkaid University, B.P. 230, Tlemcen 13000, (Algeria)

The full thermodynamic study of the absorption refrigeration units requires

the knowledge of the thermodynamic properties of the used mixture. The

present work deals with the mathematical modeling of the thermodynamic

properties of ammonia-water mixtures using various models. The presented

model covers high vapor-liquid equilibrium pressures up to 110 [bar] and

temperatures from 230 to 600 [K]. Furthermore, the calculation of the

thermodynamic properties of the ammonia-water mixtures and their pure

components was carried out. The obtained results were compared with

results given in the literature. This shows a good concordance.

Key words: ammonia-water; binary solution; thermodynamics properties;

Markel; Oldham.

1. Introduction

The mathematical models of calculation and simulation of the systems with absorption require

the knowledge of a great number of operating fluids thermodynamic properties [1]. To achieve this

goal, a great number of experimental research was carried out, where the results were presented in a

series of tables and diagrams; the known (H, x) diagram is well realized by Merkel and Bošnjaković

[2].

When calculating the performance of an ammonia-water absorption refrigeration cycle,

thermodynamic properties of the ammonia-water mixtures need to be known. These properties are

usually obtained from equations of states and other general equations [3].

The correlations for thermodynamic properties of ammonia-water mixtures found in literature,

can be divided into nine groups: cubic equations of state [4-17], virial equations of state [16, 22, 23,

25], Gibbs excess energy [7, 10, 13, 15-20, 23-32, 44-48], corresponding states method [33-38],

perturbation theory [39, 47], group contribution theory [12, 21], Leung-Griffiths model [40], Helmoltz

free energy [41] and polynomial functions [42-43].

Corresponding author. E-mail: kherris_sahraoui@yahoo.fr ; Tel/Fax: 00213 40 74 24 72

2

Different correlations for the thermodynamic properties of ammonia-water mixtures have

been used in studies of ammonia-water absorption refrigeration cycles presented in literature; they

were obtained using a library of subroutines developed by Sergio Stecco and Umberto Desideri [30].

Their correlations are based on work presented by Ziegler and Trepp [27] and El-Sayed and Tribus

[32]. Different equations are used for the vapor and liquid phases. The vapor is supposed to be an ideal

mixture of real gases while the properties of the liquid phase are corrected by a term calculated from

the Gibbs excess energy. The functions presented by Ziegler and Trepp [27] have been correlated to

experimental data for the mixture up to 240 [°C] and 52 [bar] while El-Sayed and Tribus [32] have

taken into account data up to 316 [°C] and 210 [bar]. The maximum pressure for the correlations from

Stecco and Desideri [30] has been set to 115 [bar], which is slightly above the critical pressure of

ammonia. The critical pressure and temperature of the mixture change with the mass fraction of

ammonia. Rainwater and Tillner-Roth [40] presented a model based on the so-called Leung-Griffiths

model for the ammonia-water mixture critical region vapor-liquid equilibrium. They conclude that

their model is a reasonable representation of the high-pressure phase boundary for the ammonia-water

mixture but that more experimental data would be valuable to confirm the model. Furthermore, they

mention that there are inconsistencies among the experimental data available today for the ammonia-

water mixture in the critical region.

Besides, the correlations presented and used by Xu and Goswami [29] are the same as the

correlations of Stecco and Desideri [30]. All but Park and Enick et al. [21] use different equations for

the liquid and vapor phases. Just as for the correlations of Stecco and Desideri [30] the vapor is

supposed to be an ideal mixture of real gases and the properties for the liquid phase are corrected by a

term calculated from the Gibbs excess energy for the correlations by El-Sayed and Tribus [32],

Kouremenous and Rogdakis [26] and Ibrahim and Klein [28]. Ibrahim and Klein use the equations

presented by Ziegler and Trepp [27].

However, the constants in the functions for the Gibbs excess energy have been recalculated

with experimental data at higher temperatures and pressures (up to 210 [bar] and 316 [°C]). Ziegler

and Trepp [27] have modified the correlations presented by Schulz [25]. By these modifications the

correlations are extended from a maximum pressure of 25 [bar] to 50 [bar]. Kouremenos and Rogdakis

[26] also use the equations of Ziegler and Trepp [27]. For pressures higher than 50 [bar], they use an

expression for the heat capacity given by El-Sayed and Tribus [32] to calculate enthalpy and entropy.

Nag and Gupta [31] use the Peng-Robinson equation for the vapor-liquid equilibrium and the Gibbs

free energy equations presented by Zielger and Trepp [27] for the vapor and liquid phase properties.

Park [35] uses a generalized equation of state based on the law of corresponding states to

calculate the thermodynamic properties of the mixture. Corrections for non-ideal and polar behavior

are included. Enick et al. [21] use a cubic equation of state, the Peng-Robinson equation of state, with

binary interaction parameters estimated from a group contribution model and Gibbs excess energy

equations.

Tillner-Roth and Friend [41] presented a new correlation for the ammonia-water mixture

thermodynamic properties. It is based on a fundamental equation of state for the Helmholtz free

energy and, like the correlation by Park [35]; the entire thermodynamic space of the mixture is

described by one single equation. This is formed by the fundamental equations of state of the pure

components and a mixing rule for the independent variables, reduced temperature and reduced density.

The term for the departure from non-ideal mixture behavior is fitted to the most reliable, available

3

experimental data for the ammonia-water mixture and also to data for the critical region from the

modified Leung-Griffiths model presented by Rainwater and Tillner-Roth [40].

Weber [34] has presented a model in order to estimate the second and the third virial

coefficient of the ammonia-water mixtures. Rukes and Dooley [44] have calculated the free energy of

Helmotz of the components for the derivation after the thermodynamic properties at saturation.

Many researchers have used the calculation of the free enthalpy of Gibbs to formulate the

thermodynamic properties of the ammonia-water mixtures [7, 10, 13, 15-20, 23-32, 44-48].

Moreover, M. Barhoumi and co-workers [46], for the liquid phase, have used a three constant

Margules model of the excess free enthalpy. The vapor phase is considered as a perfect mixture of real

gases, each pure gas being described by a pressure virial equation of state truncated after the third

term. Kh. Mejbri and Bellagi [47] have used three different theoretical approaches for the modelling of

thermodynamic properties of the ammonia-water mixtures. A comparison of these three methods

proves the superiority of PC-SAFT (Perturbed Chain Statistical Associating Fluid Theory) in

predicting and extrapolating the thermodynamic properties of ammonia-water system up to very high

temperatures and pressures.

Among the mentioned models, we have chosen in this study the one of Michel FEIDT [48],

which combines the Gibbs free energy method for the thermal properties and the equations calculating

the bubble point and the dew point of the mixture [32]. This method combines the advantages of the

both methods. In the same time it avoids the need to make iterations in order to have the equilibrium

conditions of phases. The suggested correlations cover equilibrium conditions of phases at high

pressures and temperatures: 230<T<600 [K] and 0.2<P<110 [bar].

This field is under the critical points of all the components, so that the determination of the

equation of state for the mixture, does not take into account features referred to the critical state field.

The equation of state does not describe the state in which the solution is in the solid-state aggregation.

The organization of the paper is as follows. The properties of working fluids of the chosen

model are given Section 2. The results of calculation of the thermodynamics properties of ammonia-

water mixtures and their pure components are presented in section 3. A comparative study of the

obtained results with the results provided by the IIR (International Institute of Refrigeration) [50] and

Dae.Wen. SUN [49] for the mixtures, and those provided by KUMAN Ražnjevic [51] and L. HAAR et

al. [52] for the pure components are carried out in the same section. Conclusions are drawn in the last

section.

2. Properties of working fluids

2.1. Pure components

2.1.1. Gibbs free energy

The fundamental equation of the Gibbs energy is given in an integral form as (Ziegler and

Trepp [27], Ibrahim and Klein [28], Xu and Yogi Goswami [29]):

0 0 0

0 0

T P T

T P T

CpG H TS CpdT V dP T dT

T (1)

The volume V and the heat capacity at constant pressure Cp for the liquid and vapor phase of

4

both pure water and pure ammonia are assumed to be empirically correlated by the following

expression (Ziegler and Trepp [27]):

2

1 2 3 4

LV a a P a T a T (2) 2

32 41 3 11 11

g cc c PRTV c

P T T T (3)

2

1 2 3

LCp b b T b T (4)

22

1 2 3 2

0

P

g VCp d d T d T T dP

T

(5)

The development of eq. (1) with the help of eqs. (2-5) leads to the following equations:

Liquid phase:

2 2 3 332,0 ,0 1 ,0 ,0 ,0 1 2 ,0

,0

2 2 2 2 23 2,0 1 3 4 ,0 ,0

( ) ( ) ( ) ln ( )2 3

( ) ( )( ) ( )2 2

L L L rr r r r r r r r r r r r r r

r

r r r r r r r r r

BB TG H T S B T T T T T T B T B T T T

T

B AT T T A A T A T P P P P

(6)

Vapor phase:

2 2 3 332,0 ,0 1 ,0 ,0 ,0 1

,0

2 232 ,0 ,0 1 ,0

,0

,0 ,0 ,0

2 33 3 4 11

,0 ,0

( ) ( ) ( ) ln2 3

( ) ( ) ln ( )2

4 3 12

g g g rr r r r r r r r r r r

r

rr r r r r r r r r

r

r r r rr r

r r r r r

DD TG H T S D T T T T T T D T

T

D PD T T T T T T T C P P

P

P P T PP PC C

T T T T T

,0

11 12

,0 ,0

3 33,0 ,04

11 11 12

,0 ,0

11

12 113

r r

r

r r rr

r r r

P T

T

P P TC P

T T T

(7)

The reduced thermodynamic properties (subscript r) are Tr=T/TB, Pr = P/PB, Gr=G/RTB,

hr=h/RTB, Sr=S/R and Vr=VPB/RTB. The values of the constants R, TB and PB are R=8.314

[kJkmol-1

K-1

], TB=100 [K] and PB=10 [bar]. Coefficients for eqs. (6-7) are given in tab. 1.

2.1.2. Thermodynamic properties

The molar specific enthalpy, entropy and volume are related to Gibbs free energy by:

2

r

r r

B r

r P

G TH RT T

T

(8)

r

r

r P

GS R

T

(9)

r

B r

B r T

RT GV

P P

(10)

5

Ammonia water

A1 3.971423.10-2

2.748796.10-2

A2 -1.790557.10-5

-1.016665.10-5

A3 -1.308905.10-2

-4.452025.10-3

A4 3.752836.10-3

8.389246.10-4

B1 1.634519.101 1.214557.101

B2 -6.508119 -1.898065

B3 1.448937 2.911966.10-1

C1 -1.049377.10-2

2.136131.10-2

C2 -8.288224 -3.169291.101

C3 -6.647257.102 -4.634611.10

4

C4 -3.045352.103 0

D1 3.673647 4.019170

D2 9.989629.10-2

-5.175550.10-2

D3 3.617622.10-2

1.951939.10-2

,0

L

rH 4.878573 21.821141

,0

g

rH 26.468879 60.965058

,0

L

rS 1.644773 5.733498

,0

g

rS 8.339026 13.453430

Tr,0 3.2252 5.0705

Pr,0 2 3

Table 1. Coefficients for eqs. (6-7)

2.2. Mixture of ammonia and water

2.2.1. Ammonia-Water liquid mixture

According to the analysis given by Ziegler and Trepp [27], the liquid mixture Gibbs function

of ammonia-water is given by the ideal solution mixture relation plus the Gibbs excess energy. This

excess energy is limited to three terms and takes into account the deviation from ideal solution

behavior.

2

1 2 31 2 1 2 1E

rG x x F F x F x

(11)

where:

5 61 1 2 3 4 2

( )r r r

r r

E EF E E P E E P T

T T (12)

11 122 7 8 9 10 2

( )r r r

r r

E EF E E P E E P T

T T (13)

6

15 163 13 14 2r

r r

E EF E E P

T T (14)

Coefficients for eqs. (12-14) are given in tab. 2. The excess specific enthalpy, entropy and

volume of the liquid mixture can be derived as eqs. (8-10).

Table 2. Coefficients for eqs. (12-14)

,

2

r x

E

r rE

B r

rP

G TH RT T

T

(15)

,r x

EE r

r P

GS R

T

(16)

,r x

EE B r

B r T

RT GV

P P

(17)

Therefore, the specific enthalpy, entropy and volume of the liquid mixture become:

3 2(1 )L L L E

m NH H OH xH x H H (18)

3 2(1 )L L L E ml

m NH H OS xS x S S S (19)

3 2(1 )L L L E

m NH H OV xV x V V (20)

where:

ln( ) (1 )ln(1 )mlS R x x x x (21)

2.2.2. Ammonia-Water vapor mixture

The vapor phase is considered as an ideal solution, so the excess Gibbs function is equal to

zero. The specific enthalpy, entropy and volume of the vapor mixture are calculated from:

3 2(1 )g g g

m NH H OH yH y H (22)

3 2(1 )g g g mg

m NH H OS yS y S S (23)

3 2(1 )g g g

m NH H OV yV y V (24)

where:

ln( ) (1 )ln(1 )mgS R y y y y (25)

E1 -41.733398 E9 0.387983

E2 0.02414 E10 0.004772

E3 6.702285 E11 -4.648107

E4 -0.011475 E12 0.836376

E5 63.608967 E13 -3.553627

E6 -62.490768 E14 0.000904

E7 1.761064 E15 24.361723

E8 0.008626 E16 -20.736547

7

3. Results and discussions

3.1. Comparison of the results

The comparison is carried out for the calculated thermodynamic properties for the mixtures

and pure components.

3.1.1. Ammonia-water mixtures

Comparison between results given by the chosen model and the experimental results provided

by IIR [50] and Dae.Wen. SUN [49] data’s of ammonia-water mixtures are presented in tab 3. The

comparison is made for the saturated pressure and saturated volume for the two states liquid and

vapor, in case of two references values of temperatures: low one equal to 283.15 [K] and high

temperature equal to 525.15 [K].

Average error [%]

T= 283.15 [K] T= 525.15 [K]

Model 1 :

IIR

Model 2 :

Dae Wen SUN

Model 1 :

IIR

Model 2 :

Dae Wen SUN

Saturated pressure 3.6 3.1

Saturated volume liquid 2.8 9.4

Saturated volume vapor 4.5 4.1

Table 3. Average error for comparative study in case of mixtures

From the tab. 3 it is clear that the average error between the calculated saturated pressure and

those given by the model 1 and 2 is up to 3.6 % for low temperatures and up to 3.1 % for high

temperatures. On the other hand, the average error in case of specific liquid volume is up to 2.8 % for

low temperatures and up to 9.4 % for high temperatures. This can be explained by the fact that most of

the elaborated models are appropriate for the vapor phase rather than for the liquid phase. Finally, the

average error for specific vapor volume is up to 4.5 % for low temperatures and up to 4.1 % for high

temperature.

3.1.2. Pure components

The chosen model allows calculating the enthalpy, latent heat, entropy and volume of

ammonia and water in case of vapor and liquid states for pure components. The results are compared

with the KUMAN Ražnjevic [51] and L. HAAR et al. [52] data’s and represented in the figs (1-8).

Figure 1. Latent heat of vaporization of ammonia

Figure 2. Latent heat of vaporization of water

The obtained variation of latent heat of vaporization for ammonia and water is compared with

KUMAN Ražnjevic and L. HAAR et al. data’s and shown in the figs (1-2) . As can be seen, the latent

heat of vaporization decreases when the temperature is increasing. However, beyond the temperature

8

value of 540 K a little difference is observed. The correlation coefficient between the results of

simulation and those of KUMAN Ražnjevic and L. HAAR et al. is equal to 0.9999, 0.9977 for

ammonia and water respectively. This coefficient is nearly the same for the other results shown in figs

(3-8).

Figure 3. Entropy variation of ammonia

Figure 4. Entropy variation of water

Figure 5. Saturated liquid volume of ammonia

Figure 6. Saturated liquid volume of water

Figure 7. Saturated vapor volume of ammonia

Figure 8. Saturated vapor volume of water

Finally, the comparative study shows a good agreement of the results of simulation with those

of the authors [51-52].

3.2. Exploring simulation results of the thermodynamic properties of ammonia-water mixtures for

the elaboration of the Merkel’s and Oldham’s diagrams

3.2.1. Merkel’s diagram

Merkel’s diagram allows doing a full study of the absorption machine, as it provides the heat

balances of the various parts of the circuit by direct reading of enthalpy differences. As shown in fig.9,

the x-axis is dedicated to the concentration of the liquid phase and the y-axis to the enthalpy.

It includes at its lower part, networks of isobars and isotherms, as well as curves of vapor-

liquid equilibrium at equal concentration. At the upper part, the reference curves allow to establish the

characteristics of the vapor phase, starting from a determined point of equilibrium in the lower part.

Figure 9. Merkel’s diagram

3.2.2. Oldham’s diagram

This diagram is the most used and most convenient for the absorption machine study. As

shown in fig.10, the x-axis is scaled in (1/T) and the y-axis in (Log P). In this system of coordinates,

the curves reflecting the balance of the binary system in the vapor as well as in the liquid phase are

approximately straight lines. The straight line with 100% of concentration corresponds to the vapor-

liquid equilibrium of pure ammonia, and the one with 0% of concentration correspond to vapor-liquid

equilibrium of pure water.

Figure 10. Oldham’s diagram

4. Conclusion

The ammonia-water mixture was the principal binary couple used in the absorption

refrigerating machines for several years. As in many studies, this one is conducted on vapor-liquid

balance and the thermo-physical properties of mixture. For that the following conclusions were drawn:

9

A method that combines the Gibbs free energy method for mixture properties and

bubble and dew point temperature equations for phase equilibrium is used. This

method combines the advantages of the two and avoids the need for iterations for

phase equilibrium;

The proposed correlations cover high vapor-liquid equilibrium pressures and

temperatures;

It is necessary to note the good agreement of the calculated of the thermodynamic

properties for the mixture from the chosen model with the IIR and Dae. Wen. SUN

data’s;

The comparative analysis for the calculated enthalpy, latent heat, entropy and volume

of ammonia and water in case of vapor and liquid states for pure components with

KUMAN Ražnjevic and L. HAAR et al. data’s shows a good agreement. This is

confirmed by the values of the correlation coefficient equal to 0.999 for ammonia and

water.

Nomenclature

H molar enthalpy, [kJkmol-1

]

V molar volume, [m3kmol

-1]

S molar Entropy, [kJkmol-1

K-1

]

T temperature, [°C, K]

P pressure, [bar]

G Gibbs free energy, [kJkmol-1

]

x liquid ammonia mole fraction, [kmol of NH3 kmol-1

of liquid mixture]

y vapor ammonia mole fraction, [kmol of NH3 kmol-1

of vapor mixture]

R molar gas constant, [kJkmol-1

K-1

]

Cp heat capacity at constant pressure, [kJkg-1

K-1

]

a, b, c, d coefficients, []

A, B, C, D, E, F dimensionless coefficients, []

r correlation coefficient, []

Subscripts Superscripts

c critical E excess

NH3 ammonia ml liquid mixture

H2O water mg vapor mixture

r reduced L liquid

0 reference state g vapor

m mixture

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13

Figure 1. Latent heat of vaporization of ammonia

Figure 2. Latent heat of vaporization of water

Figure 3. Entropy variation of ammonia

14

Figure 4. Entropy variation of water

Figure 5. Saturated liquid volume of ammonia

Figure 6. Saturated liquid volume of water

15

Figure 7. Saturated vapor volume of ammonia

Figure 8. Saturated vapor volume of water

16

Figure 9. Merkel’s diagram

17

Figure 10. Oldham’s diagram